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In this paper, the authors raise a question on the theoretical foundation of option implied risk neutral density. They prove that given any number of options, there exist numerous risk neutral densities which are piecewise constant, have only two values, either a lower bound or an upper bound on the true risk neutral density, and price all these options correctly. Similar results are proved with respect to the true risk neutral density's derivatives. These results show how difficult it is to ensure that the risk neutral density they extract from option prices is the true one and how large estimation errors can be.
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